[flopunctro]

> a slight push in the direction of the sun

That's not how orbital dynamics work; a slight push in any direction won't change the orbit enough for that thing to not be a problem anymore.

To make something fall into the sun, one needs quite a bit of delta-vee, which means energy. However, if the thing isn't in any hurry (and I imagine waste isn't), one can simply attach a solar sail with a tiny computer for steering it, and let it brake for the next few thousand years.

(Kerbal Space Program should be part of school curricula)